In "Deriving Auslander's Formula (Theorem 2.2)", I'm trying to construct a right adjoint to inclusion functor $\mathsf{i:eff~C \to mod~C}$. I constructed it as follows:
The functor $\mathsf{j:mod~C \to eff~C}$ (right adjoint of $\mathsf{i}$) sends each object $f$ to itself if $f$ is in $\mathsf{eff~C}$, and otherwise it sends $f$ $($with presentation $(-,A) \to (-,B) \to f \to 0$$)$ to $\mathsf{coker}((-,B) \to (-,coker(A \to B)))$.Now, my problem is: "If $Hom_{\mathsf{C}}(c,-)$ is exact, the latter assignment (above) sends each object to zero object in $\mathsf{eff~C}$. In this case, zero object in $\mathsf{eff~C}$ has many presentations." Is abelianness of $\mathsf{C}$ sufficient for $Hom_{\mathsf{C}}(c,-)$ to be exact? If yes, how can I resolve this difficulty?
Notes and definitions:
$\mathsf{C}$: an abelian category
$\mathsf{(C^{op},Ab)}$: the category of functors from $\mathsf{C^{op}}$ to $\mathsf{Ab}$
Finitely presented object: An object $F$ in $\mathsf{(C^{op},Ab)}$ is called finitely presented if it has an exact presentation $(-,A) \to (-,B) \to F \to 0$
$\mathsf{mod~C}$ : the full subcategory of $\mathsf{(C^{op},Ab)}$ consisting of finitely presented objects
$\mathsf{eff~C}$: the full subcategory of $\mathsf{mod~C}$ consisting of objects $F$ (in definition of "finitely presented object") with $A \to B$ an epimorphism in $\mathsf{C}$